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Frederic Latremoliere

Publications and source records attributed to Frederic Latremoliere.

54 records · Page 3Linked to original sources
arXiv2013

Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach

Quantum tori are limits of finite dimensional C*-algebras for the quantum Gromov-Hausdorff propinquity, a metric defined by the author as a strengthening of Rieffel's quantum Gromov-Hausdorff designed to retain the C*-algebraic structure. In this paper, we propose a proof of the continuity of the family of quantum and fuzzy tori which relies on explicit representations of the C*-algebras rather than on more abstract arguments, in a manner which takes full advantage of the notion of bridge defining the quantum propinquity.

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arXiv2013

Noncommutative solenoids and their projective modules

Let p be prime. A noncommutative p-solenoid is the C*-algebra of Z[1/p] x Z[1/p] twisted by a multiplier of that group, where Z[1/p] is the additive subgroup of the field Q of rational numbers whose denominators are powers of p. In this paper, we survey our classification of these C*-algebras up to *-isomorphism in terms of the multipliers on Z[1/p], using techniques from noncommutative topology. Our work relies in part on writing these C*-algebras as direct limits of rotation algebras, i.e. twisted group C*-algebras of the group Z^2 thereby providing a mean for computing the K-theory of the noncommutative solenoids, as well as the range of the trace on the K_0 groups. We also establish a necessary and sufficient condition for the simplicity of the noncommutative solenoids. Then, using the computation of the trace on K_0, we discuss two different ways of constructing projective modules over the noncommutative solenoids.

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arXiv2012

Quantum Locally Compact Metric Spaces

We introduce the notion of a quantum locally compact metric space, which is the noncommutative analogue of a locally compact metric space, and generalize to the nonunital setting the notion of quantum metric spaces introduced by Rieffel. We then provide several examples of such structures, including the Moyal plane, as well as compact quantum metric spaces and locally compact metric spaces. This paper provides an answer to the question raised in the literature about the proper notion of a quantum metric space in the nonunital setup and offers important insights into noncommutative geometry for non compact quantum spaces.

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arXiv2012

Classification of Noncommutative Domain Algebras

Noncommutative domain algebras are noncommutative analogues of the algebras of holomorphic functions on domains of $\C^n$ defined by holomorphic polynomials, and they generalize the noncommutative Hardy algebras. We present here a complete classification of these algebras based upon techniques inspired by multivariate complex analysis, and more specifically the classification of domains in hermitian spaces up to biholomorphic equivalence.

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arXiv2011

Noncommutative Solenoids

A noncommutative solenoid is the C*-algebra $C^\ast(\Q_N^2,σ)$ where $\Q_N$ is the group of the $N$-adic rationals twisted and $σ$ is a multiplier of $\Q_N^2$. In this paper, we use techniques from noncommutative topology to classify these C*-algebras up to *-isomorphism in terms of the multipliers of $\Q_N^2$. We also establish a necessary and sufficient condition for simplicity of noncommutative solenoids, compute their K-theory and show that the $K_0$ groups of noncommutative solenoids are given by the extensions of $\Z$ by $\Q_N$. We give a concrete description of non-simple noncommutative solenoids as bundle of matrices over solenoid groups, and we show that irrational noncommutative solenoids are real rank zero AT C*-algebras.

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arXiv2011

Symmetry in the Cuntz Algebra on two generators

We investigate the structure of the automorphism of $\mathcal{O}_{2}$ which exchanges the two canonical isometries. Our main observation is that the fixed point C*-subalgebra for this action is isomorphic to $\mathcal{O}_{2}$ and we detail the relationship between the crossed-product and fixed point subalgebra.

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arXiv2010

Isomorphisms of noncommutative domain algebras II

This paper extends the results of the previous work of the authors on the classification on noncommutative domain algebras up to completely isometric isomorphism. Using Sunada's classification of Reinhardt domains in $C^n$, we show that aspherical noncommutative domain algebras are isomorphic if and only if their defining symbols are equivalent, in the sense that one can be obtained from the other via permutation and scaling of the free variables. Our result also shows that the automorphism groups of aspherical noncommutative domain algebras consists of a subgroup of some finite dimensional unitary group. We conclude by illustrating how our methods can be used to extend to noncommutative domain algebras some results from analysis in $C^n$ with the example of Cartan's lemma.

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arXiv2010

Ergodic Actions of Convergent Fuchsian groups on quotients of the noncommutative Hardy algebras

We establish that particular quotients of the non-commutative Hardy algebras carry ergodic actions of convergent discrete subgroups of the group $\operatorname*{SU}(n,1)$ of automorphisms of the unit ball in $\mathbb{C}% ^{n}$. To do so, we provide a mean to compute the spectra of quotients of noncommutative Hardy algebra and characterize their automorphisms in term of biholomorphic maps of the unit ball in $\mathbb{C}^{n}$.

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arXiv2010

Irreducible Representations of C*-crossed products by Finite Groups

We describe the structure of the irreducible representations of crossed products of unital C*-algebras by actions of finite groups in terms of irreducible representations of the C*-algebras on which the groups act. We then apply this description to derive a characterization of irreducible representations of crossed-products by finite cyclic groups in terms of representations of the C*-algebra and its fixed point subalgebra. These results are applied to crossed-products by the permutation group on three elements and illustrated by various examples.

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arXiv2009

C*-Algebraic Characterization of Bounded Orbit Injection Equivalence for Minimal Free Cantor Systems

Bounded orbit injection equivalence is an equivalence relation defined on minimal free Cantor systems which is a candidate to generalize flip Kakutani equivalence to actions of the Abelian free groups on more than one generator. This paper characterizes bounded orbit injection equivalence in terms of a mild strengthening of Rieffel-Morita equivalence of the associated C*-crossed-product algebras. Moreover, we construct an ordered group which is an invariant for bounded orbit injection equivalence, and does not agrees with the K\_0 group of the associated C*-crossed-product in general. This new invariant allows us to find sufficient conditions to strengthen bounded orbit injection equivalence to orbit equivalence and strong orbit equivalence.

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arXiv2009

Boolean Inner product Spaces and Boolean Matrices

This article discusses the concept of Boolean spaces endowed with a Boolean valued inner product and their matrices. A natural inner product structure for the space of Boolean n-tuples is introduced. Stochastic boolean vectors and stochastic and unitary Boolean matrices are studied. A dimension theorem for orthonormal bases of a Boolean space is proven. We characterize the invariant stochastic Boolean vectors for a Boolean stochastic matrix and show that they can be used to reduce a unitary matrix. Finally, we obtain a result on powers of stochastic and unitary matrices.

math.RA↗
arXiv2009

Isomorphisms of Non-Commutative Domain Algebras

Noncommutative domain algebras were introduced by Popescu as the non-selfadjoint operator algebras generated by weighted shifts on the Full Fock space. This paper uses results from several complex variables to classify many noncommutative domain algebras, and it uses results from operator theory to obtain new bounded domains in hermitian spaces with non-compact automorphic group.

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arXiv2007

Crossed-product C*-algebras for conformal automorphisms of the disk

We study the C*-algebra crossed-product of the closed unit disk by the action of one of its conformal automorphisms. After classifying the conformal automorphisms up to topological conjugacy, we investigate, for each class, the irreducible representations of the full C*-crossed-products, and derive their spectrum and a complete desciption of the algebras.

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arXiv2006

The C*-algebra of symmetric words in two universal unitaries

We compute the K-theory of the C*-algebra of symmetric words in two universal unitaries. This algebra is the fixed point C*-algebra for the order-two automorphism of the full C*-algebra of the free group on two generators which switches the generators. Our calculations relate the K -theory of this C*-algebra to the K-theory of the associated C*-crossed-product by Z_2.

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arXiv2006

C*-crossed-products by an order-two automorphism

We describe the representation theory of C*-crossed-products of a unital C*-algebra A by the cyclic group of order 2. We prove that there are two main types of irreducible representations for the crossed-product: those whose restriction to A is irreducible and those who are the sum of two unitarily unequivalent representations of A. We characterize each class in term of the restriction of the representations to the fixed point C*-subalgebra of A. We apply our results to compute the K-theory of several crossed-products of the free group on two generators.

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arXiv2005

Approximation of Quantum Tori by Finite Quantum Tori for the Quantum Gromov-Hausdorff Distance

The statement that one can approximate a quantum torus by some twisted convolution C*-algebra of a (finite) quotient of Z^d can be found in the physics literature dealing with quantum field theory and M-theory. In this paper, we show that indeed the quantum tori are limits of such finite dimensional C*-algebras for the quantum Gromov-Hausdorff distance for compact quantum metric spaces introduced by Rieffel, when one chooses the right metric structures. The proof uses a mild extension of a result by Ramazan on continuous fields of C*-algebras obtained from groupoid C*-algebras, and extends on Rieffel's work on the continuity of the quantum tori for the quantum Gromov-Hausdorff distance. We also show that one can collapse a quantum torus to a lower dimensional quantum torus.

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arXiv2005

Bounded-Lipschitz Distances on the State Space of a C*-algebra

Metric noncommutative geometry, initiated by Alain Connes, has known some great recent developments under the impulsion of Rieffel and the introduction of the category of compact quantum metric spaces topologized thanks to the quantum Rieffel-Gromov-Hausdorff distance. In this paper, we undertake the first step to generalize such results and constructions to locally compact quantum metric spaces. Our present work shows how to generalize the construction of the bounded-Lipschitz metric on the state space of a C*-algebra which need not be unital, such that the topology induced by this distance on the state space is the weak* topology. In doing so we obtain some results on a state space picture of the strict topology of a C*-algebra.

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